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Article Dans Une Revue Bulletin of the London Mathematical Society Année : 2018

Extensions between functors from free groups

Résumé

Motivated in part by the study of the stable homology of automorphism groups of free groups, we consider cohomological calculations in the category $\mathcal{F}(\textbf{gr})$ of functors from finitely generated free groups to abelian groups. In particular, we compute the groups $Ext^*_{\mathcal{F}(\textbf{gr})}(T^n \circ \mathfrak{a}, T^m \circ \mathfrak{a})$ where $\mathfrak{a}$ is the abelianization functor and $T^n$ is the n-th tensor power functor for abelian groups. These groups are shown to be non-zero if and only if $*=m-n \geq 0$ and $Ext^{m-n}_{\mathcal{F}(\textbf{gr})}(T^n \circ \mathfrak{a}, T^m \circ \mathfrak{a})=\mathbb{Z}[Surj(m,n)]$ where $Surj(m,n)$ is the set of surjections from a set having $m$ elements to a set having $n$ elements. We make explicit the action of symmetric groups on these groups and the Yoneda and external products. We deduce from these computations those of rational Ext-groups for functors of the form $F \circ \mathfrak{a}$ where $F$ is a symmetric or an exterior power functor. Combining these computations with a recent result of Djament we obtain explicit computations of stable homology of automorphism groups of free groups with coefficients given by particular contravariant functors.
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Dates et versions

hal-01224436 , version 1 (05-11-2015)
hal-01224436 , version 2 (11-04-2016)
hal-01224436 , version 3 (20-09-2016)

Identifiants

Citer

Christine Vespa. Extensions between functors from free groups. Bulletin of the London Mathematical Society, 2018, 50 (3), pp.401-419. ⟨10.1112/blms.12091⟩. ⟨hal-01224436v3⟩
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